Comparison of \(P\)-convexity, \(O\)-convexity and other geometrical properties
A real Banach space \(X\) is \(P(n,\epsilon)\)-convex, \( n \geq 2\), \(\epsilon > 0\), if, for every \(x_1, \dots,x_n\) in the closed unit ball of \(X\), there exist \(i \neq j\) such that \(\|x_i - x_j\|\;\leq 2 -\epsilon\). \(X\) is \(P(n)\)-convex if it is \(P(n,\epsilon)\)-convex for some \(\epsilon > 0\), and \(X\) is \(P\)-convex if it is \(P(n)\)-convex for some \(n \in N\). A real Banach space \(X\) is \(O(n,2\epsilon)\)-convex, \( n \geq 2\), \(\epsilon > 0\), if, for every \(x_1,\dots,x_n\) in the closed unit ball of \(X\), there exist \(i \neq j\) such that \(\min (\|x_i + x_j\|, \|x_i - x_j\|) \leq 2 (1- \epsilon)\). \(X\) is \(O(n)\)-convex if it is \(O(n, 2\epsilon)\)-convex for some \(\epsilon > 1\), and it is \(O\)-convex if it is \(O(n)\)-convex for some \(n\). Some properties of these spaces are discussed. Fixed point theorems are proved and some examples are provided.
scientific article; zbMATH DE number 6096070
scientific article